ANALYSIS OF VARIANCE AND COVARIANCEsame, then the stimulus didn’t ‘‘make a differ-
ence’’ in the outcome variable. If the means are
different, then, because of random assignment to
groups and standardization of conditions, it is
assumed that the stimulus caused the difference.
Of course, it is necessary to establish some guide-
lines for interpreting the size of the difference in
order to draw conclusions regarding the strength
of the effect of the experimental stimulus. The
criterion used by analysis of variance is the amount
of random variation that exists in the scores within
each group. For example, in a three-group com-
parison, the group mean scores on some outcome
measure may be 3, 6, and 9 on a ten-point scale.
The meaningfulness of these differences can only
be assessed if we know something about the varia-
tion of scores in the three groups. If everyone in
group one had scores between 2 and 4, everyone
in group two had scores between 5 and 7, and
everyone in group three had scores between 8 and
10, then in every instance the variation within each
group is low and there is no overlap across the
within group distributions. As a result, whenever
the outcome score of an individual in one group is
compared to the score of an individual in a differ-
ent group, the result of the comparison will be very
similar to the respective group mean score com-
parison and the conclusions about who scores
higher or lower will be the same as that reached
when the means were compared. As a result, a
great deal of confidence would be placed in the
conclusion that group membership makes a differ-
ence in one’s outcome score. If, on the other hand,
there were several individuals in each group who
scored as low or as high as individuals in other
groups—a condition of high variability in scores—
then comparisons of these subjects’ scores would
lead to a conclusion opposite to that represented
by the mean comparisons. For example, if group
one scores varied between 1 and 5 and group two
scores varied between 3 and 10, then in some cases
individuals in group one scored higher than indi-
viduals in group two (e.g., 5 vs. 3) even though the
mean comparisons show the opposite trend (e.g.,
3 vs. 6). As a result, less confidence would be
placed in the differences between means.
Although analysis of variance results reflect a
comparison of group means, conceptually and
computationally this procedure is best understood
through a framework of explained variance. Rath-
er than asking ‘‘How much difference is there
between group means?,’’ the question becomes
‘‘How much of the variation in subjects’ scores on
the outcome measure can be explained or ac-
counted for by the fact that subjects were exposed
to different treatments or stimuli?’’ To the extent
that the experimental stimulus has an effect (i.e.,
group mean differences exist), individual scores
should differ from one another because some
have been exposed to the stimulus and others have
not. Of course, individuals will differ from one
another for other reasons as well, so the procedure
involves a comparison of how much of the total
variation in scores is due to the stimulus effect (i.e.,
group differences) and how much is due to extra-
neous factors. Thus, the total variation in outcome
scores is ‘‘decomposed’’ into two elements: varia-
tion due to the fact that individuals in the different
groups were exposed to different conditions, ex-
periences, or stimuli (explained variance); and
variation due to random or chance processes (er-
ror variance). Random or chance sources of varia-
tion in outcome scores can be such things as
measurement error or other causal factors that are
randomly distributed across groups through the
randomization process. The extent to which varia-
tion is due to group differences rather than these
extraneous factors is an indication of the effect of
the stimulus on the outcome measure. It is this
type of comparison of components of variance
that provides the foundation for analysis of variance.BASIC CONCEPTS AND PROCEDURESThe central concept in analysis of variance is that
of variance. Simply put, variance is the amount of
difference in scores on some variable across sub-
jects. For example, one might be interested in the
effect of different school environments on the self-
esteem of seventh graders. To examine this effect,
random samples of students from different school
environments could be selected and given ques-
tionnaires about their self-esteem. The extent to
which the students’ self-esteem scores differ from
each other both within and across groups is an
example of the variance.Variance can be measured in a number of
ways. For example, simply stating the range of
scores conveys the degree of variation. Statistical-
ly, the most useful measures of variation are based
on the notion of the sums of squares. The sums of
squares is obtained by first characterizing a sample